modeling multiple critical depths

WATER RESOURCES BULLETINVOL. 30, NO.1 AMERICAN WATER RESOURCES ASSOCIATION FEBRUARY 1994

MODELING MULTIPLE CRITICAL DEPTHS 1

INTRODUCTION

Robert G. Traver2

The purpose of this work is to investigate occur-rences of multiple critical depths in computationalmodels of open channel flow, to review their charac-teristics, and to determine means of identification andstrategies to manage the effect of their occurrences.Current water resource planning typically incorpo-rates a mathematical model depiction of the systemunder analysis. This model is exercised to predicteffects of various design changes to the river struc-ture, rainfall loadings, or other changes to the overallriver environment. Current practice includes one-dimensional steady and unsteady flow models such asHEC-2 (U.S. Army Corps of Engineers, 1990) andDAMBRK (Fread, 1988).

Knowledge of critical depth is essential for one-dimensional water surface models. From a hydraulicviewpoint, critical depth indicates a change fromgravity driven subcritical flow to inertial drivensupercritical flow, as well as a change in direction ofthe wave characteristics. When viewed as a compo-nent of a numerical model, critical depth may be used

to represent a boundary condition, a limit to the solu-tion space, or a point of decision used to evaluate theflow regime. For some flow models and situations,knowledge of critical depth is required at every crosssection for every time step (U.S. Army Corps of Engi-neers, 1990; Fread, 1988; Prayer, 1988). Traditionally,multiple critical depths are not accounted for in pre-sent models as they can be considered a computation-al hindrance of the one-dimensional assumption. Thisomission needs to be corrected as the unanticipatedoccurrence of an 'extra' critical depth may produceunreliable and inaccurate results.

GOVERNING EQUATIONS

Critical depth has been defined as the point of min-imum specific energy (Chow, 1959). This singular defi-nition must be revised for mathematical one-dimensional models because Blalock and Sturm(1981), Chaudry and Bhallamudi (1988), and severalothers have reported the occurrence of multiple criti-cal depths.

The critical depth equation (Equation 2) is found bysetting the first derivative of specific energy (Se)(Equation 1) with respect to depth equal to zero(Chow, 1959).

2aQ 1Se=Y+ 22gA

1Paper No. 93088 of the Water Resources Bulletin. Discussions are open until October 1, 1994.2Assistant Professor, Department of Civil Engineering, Vilanova University, Vilanova, Pennsylvania 19085.

85 WATER RESOURCES BULLETIN

ABSTRAC1 This paper explores the occurrence of multiple criticaldepths in one-dimensional computational models of open channelsystems. The mathematical formulation is reviewed, includingexamination of the number of possible roots by Descartes' Rule.Governing equations and dependent variables are scrutinized usingtwo compound cross sections. Occurrence tendencies are reviewedfor singular channels. Critical flow is introduced as a tool to deter-mine the existence and location of computationally based multiplecritical depths. A strategy to manage multiple critical depths inexisting one-dimensional steady or unsteady models is proposed.(KEY TERMS: critical depth; open channel hydraulics; modeling;water resources planning; simulation.)

dS

dyaQ2 dA0=1— .——+gA

Qda2d

2gA

(2)

TI (2/3)K=-AR (4)

with Q representing the channel flow, y the depth, Athe channel area, g the gravitational coefficient, a theenergy correction coefficient, R the hydraulic radius,K the conveyance, and i, a constant equaling 1 for SIor 1.486 for English units. Note that conveyance isdetermined through use of separate channel and over-bank areas as required by Manning's relationship.

ROOTS OF THE EQUATIONS

Before applying any equation, it is necessary todetermine the number of possible roots. Equation (1)appears to be cubic if a is assumed dimensionless,area is represented as a function of depth, and theequation is rearranged. This is only true for rectangu-lar channels. Figure 1 shows that five differentdepths all represent one specific energy (15 lengthunits) for trapezoidal channels. This is because thespecific energy equation for trapezoidal channels is afifth order equation, due to the added depth term inthe area representation. Descartes' rule of signs(Southworth and Deleeuw, 1965) indicates that fortrapezoidal channels, two positive and three negativeroots exist. The same analysis for rectangular chan-nels identifies two positive and one negative roots asconfirmed by Chaudry (1993). As Equation (1) issolved for depth, only positive values are possible,eliminating the necessity to consider negative roots.Critical depth therefore represents a singular multi-ple positive root and minimum specific energy forboth trapezoidal and rectangular channels. As mostmodels use an X-Y coordinate structure that mimicsincremental trapezoidal and rectangular shapes, it istempting to state that only one critical depth is possi-ble. This is incorrect for one-dimensional mathemati-cal models as multiple critical depths have beenshown to exist for compound singular channels(Blalock and Storm, 1981; Chaudry and Bhallamudi,1988; Prayer, 1993).

Figure 1. Depthvs. Specific Energy — Trapezoidal Channel.

COMPOUND CHANNEL GEOMETRY

Previous investigators (Blalock and Storm, 1981;Chaudry and Bhallamudi, 1988) have employed twocompound channels (Figure 2) for the focus of theirwork. Inspection of Figure 2 reveals that while bothcross sections have the same singular main channel,the overbanks are different. Cross section A has awide, flat overbank that is instantaneously engagedas the flow goes out of bank. The overbank of crosssection B is sloped at a one degree angle, graduallyincluding the overbank as the water rises. These ten-dencies are reflected in plots of depth versus area anda for the two cross sections (Figure 3). Both plots areidentical below the six foot mark, with section B dis-playing a gradual change in both area and a as thedepth rises into the overbank. This is not true forcross section A. As the flow rises out of bank, theentire overbank is engaged instantaneously. This pro-duces a severe change in slope for both area and a at

WATER RESOURCES BULLETIN 86

(3 2IK./A.— I I

—3 —2K/A

Traver

(3)

.C

0

—2a —15 —1U —5 0 5 10 15

Specific Energy (Se) -- Length

20 25 30

>

aa)

Figure 2. Compound Channels A and B.

Modeling Multiple Critical Depths

_(adA ida3dv 2dygA ' 2gA

Plotting critical flow versus depth (Figure 4) forcross sections A and B expands upon the work of pre-vious authors. For each cross section, the critical con-dition can be directly developed from the channelgeometry, and thus can be explored prior to using asteady or unsteady water surface profile model. Fig-ure 4 confirms that, computationally, multiple criticaldepths exist. Both cross sections have the same rela-tionship between critical depth and flow when theflow is confined to the channel. Once out of channel,the critical flow-depth relationship for cross section Adrops sharply downward, while cross section B contin-ues smoothly upward. Clearly, the criteria for definingcritical depth must be revised as both local specificenergy minima and maxima satisfy the critical depthequation (Equation 2) and Froude Number criteria.The jagged appearance of the curve is due to the dis-cretization interval used to calculate the derivatives,and is not a hydraulic property.

8500

7500a)

()

L6500

4 --550000

C—)

4500

3500

Figure 4. Critical Flow vs. Depth.

CRITICAL DEPTH INCOMPOUND CHANNELS

The channel geometry influence on critical depthcan be better explored by rearranging Equation (2).By factoring out the flow squared term, a relationshipbetween flow and critical depth results (Equation 5).As the flow term corresponds to the critical condition,it is termed critical flow (Qc) (Traver, 1993).

As shown by Figure 4, it is possible to have threedistinct depths that satisfy the critical depth equationfor the same flow. This would easily confuse existingflow models, causing nonconvergence or a false solu-tion. For 5500 cfs and cross section A, critical depthsoccur at approximately 5.55, 6.05, and 6.90 feet, arange of 1.35 feet. Without clear guidance, any ofthese depths could be chosen by present models andused as a boundary, decision point, or solution limit.This is particularly troublesome as it is possible that


the six-foot mark. The rate of change of conveyance isalso greater for cross section A at the six foot mark,but is much more gradual than the change in area.

—.5

(5)

Depth (y) —— Ft

Figure 3. Area and a vs. Depth.

the local maximum (6.05 feet) could be confused forthe point of minimum specific energy. What is evenmore disturbing is that four depths represent thesame specific energy. Figure 5 focuses on cross sectionA by adding a range of specific energy plots. For 5000cfs, a specific energy of 7.75 feet can be mathematical-ly represented by depths of 4.6, 5.9, 6.25, or 7.2 feet. Ifthe lower 'critical depth' is chosen by a backwatersteady flow model, any of the upper three depthscould be used to represent a specific energy of 7.75feet, a possible range in error of 1.3 feet. Multiple crit-ical depths can be a significant source of error, andare currently affecting results from models used instandard practice.

The drastic change in area for cross section A coin-cides with the local specific energy maximum asshown in Figures 3, 4, and 5. The initial downturn ofthe critical flow graph starts at the top of bank andreverses itself after a small depth is realized in theoverbank. Figure 6 plots critical depth versus criticalslope. Critical slope is obtained from Mannings rela-tionship using critical flow (Equation 5) and con-veyance (Equation 4) (Chow, 1959). As critical flowand conveyance are treated solely as a function ofdepth, so is critical slope. Again, a sharp discontinuityoccurs for cross section A immediately upon breachingthe flood plain, and again no such discontinuityoccurs for section B. When examining the frictionslopes related to the 5500 cfs three 'critical' depths,the slopes differ substantially, from an in-channelfriction slope of 0.00825 to 0.0025 ft/ft in the over-bank. Note the reversed friction slope gradients in theoverbank when comparing the two cross sections.

Traver

CAUSES OF MULTIPLE CRITICAL DEPTHS

The source of computationally derived multiplecritical depths is clearly due to the large change inarea as the flow leaves the channel. As our previousexamination of the order of the specific energy equa-tion returned only two possible roots (or one criticaldepth), for multiple critical depths to appear, theorder of the equation must increase by a factor of two.This can only be caused by one of the equation vari-ables. By examining compound channels with andwithout the a term, it can be extrapolated that the aterm is not the source as it is dimensionless, and itcan be shown to eliminate some multiple criticaldepth occurrences. The only other factor is the discon-tinuity in the area term. When examining criticalslope, critical flow, and specific energy plots, the dis-continuity always occurs at the local maximum of thespecific energy equation, at the point where flow goesout of bank. No multiple critical depths or discontinu-ities occurred for cross section B as the change inarea, as the flow leaves the channel is gradual. In allcases, when flow is contained within the main chan-nel, both cross sections return identical results, as thegeometric representation is the same.

Before attempting to model critical depth, it isnecessary to determine whether the multiple occur-rences are a natural hy.draulic phenomenon or simplyan aberration of our numeric representation. One-dimensional modeling assumes a constant water sur-face elevation across the channel. Specific energy iscorrected using the a term for varying velocity head.Both the channel and the overbank flows are assumedto be normal to the orientation of the overbank, and


0.008

1 0.007

20.005

0U

C

(I)

0)a)CU

U

Ua)

(i7

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11Critical Depth (yc) —— Ft

Figure 6. Critical Slope vs. Critical Depth.

U

00

01..

00

U

Figure 5. Specific Energy vs. Depth-Channel A.


have the same friction slope. For cross section A, asthe depth of flow changes from 6.0 feet to 6.5 feet, thecritical flow drops from 6300 to 4400 cfs, and the criti-cal slope drops from 0.008 to 0.0025 ftift. The con-veyance barely changes at all over this range, but thearea more than doubles and the alpha term morethan triples. This occurrence of multiple criticaldepths reflects the fact that the current one-dimen-sional mathematical depiction is not sufficient tomodel the physical processes through the channel tooverbank transition as theorized by Blalock andSturm (1983). For a depth of 0.1 foot in the overbank,the relatively low velocity overbank flows areassumed to have the same energy gradient and direc-tion as the high velocity channel flow. At this lowdepth, the overbank flow velocity in reality is negligi-ble, and can be assumed to be directed into or out ofthe main channel. This is supported by the correlationof the local specific energy maximum to the first datapoint out of channel. Note that cross section B, whichdoes not have multiple critical depths, differs fromcross section A by only a 1 percent cross slope in theoverbank.

MODELING AND MULTIPLECRITICAL DEPTHS

There is some good news for the hydraulic modeler.First, it is difficult to create a situation where thereare multiple critical depths as long as the formulationof conveyance and a is correct. A large sudden changein area is required. The U.S. Army Corps of EngineersHydrologic Engineering Center has found multiplecritical depths in cross sections without a clearly sin-gular main channel, and in cross sections with levees(G. W. Brunner, personal communication). In thisstudy, it was impossible to create any situation wheremore than two critical depths occurred for compoundchannels with a singular low flow channel. Second, asthe large change in area corresponds to the local max-imum, it separates the cross section into two zones,each with a single critical depth. The local maximumspecific energy is not considered 'critical' as it is notminimum condition. Third, as the critical flow equa-tion (Equation (5) is solely dependent upon channelgeometry, it is relatively simple to determine whethermultiple critical depths occur prior to applying awater surface profile model.

The first step in modeling critical depth is to deter-mine whether multiple occurrences exist, and at whatlocations. Chaudry and Bhallamudi (1988) developeda procedure to determine possible critical depths forsymmetric compound channels by relating flow,widths of the overbank and channel, depth of the

channel, and a ratio of channel and overbank Man-ning n. The critical flow equation (Equation 5) pro-vides another alternative for nonsymmetricalchannels. Figures 4 and 5 show that the existence ofmultiple critical depths is contingent upon the occur-rence of a negative derivative of critical flow withrespect to depth. A search technique can be employedto locate this negative derivative, thereby confirmingor precluding their existence. This technique can beincluded within a geometric model as a precursor torunning a full steady or unsteady model. As thechange in slope represents the local specific energymaximum, it separates the channel into zones, eachcontaining a singular critical depth. A combinationNewton Raphson/Bisection routine can then beapplied to each zone, using the local maximum as aboundary (Traver, 1988). The number of local maximafound determines the number of possible criticaldepths.

After locating the critical depths, a practical proce-dure is needed to decide which depth is appropriatefor the river section being modeled. This is extremelydifficult due to the questions that have been raisedearlier regarding the applicability of one-dimensionalanalysis as flow rises out of bank. Current modelsneed to include such a procedure, as unaccounted formultiple critical depth occurrences may falsely por-tray the flow event.

Several 'truths' can be ascertained from Figures 4,5, and 6. First, if the depth is above the higher orbelow the lower critical depth, the flow regime isknown. Second, as both channels have identicalcurves below the top of bank, this portion has to beconsidered correct. Third, one of the two specific ener-gy minima is equal to or lower than the other. Fourth,there is reason to question the applicability of the onedimensional assumption for small depths within theoverbanks, whether due to mixed flow (Blalock andSturm, 1983) or the directional component of the over-bank velocities. It is not reasonable that the criticalflow falls by 1500 cfs as the depth rises by 0.1 feet.Fifth, the friction slope for the lower critical depth ismuch higher than for the upper critical depth, bring-ing into question the time sequence of the modeledevent and the surrounding river structure. This isimportant for both steady and unsteady models whenmodeling flow leaving or returning to the mainchannel. From these 'truths,' a one-dimensional proce-dure is proposed for singular main channels withoutlevees using two criteria: (1) which of the two criticaldepths is lower, and (2) whether the flow depth is ris-ing or falling. This procedure is envisioned to avoidsolving for depths corresponding to areas where theone-dimensional assumption is known to be incorrect,and to maintain conservation of energy. Figures 7and 8 present this method.


Figure 7. Multiple Critical Depth Procedure — FallingDepth.

Figure 7 presents the procedure used for fallingelevations, where the momentum transfer is into thechannel. For the specific energy curve of 5500 cfs, thecritical specific energy in the overbank is lower thanthat of the channel. It is proposed to use the depthwith the lower specific energy as the boundary sepa-rating super- and sub-critical flows. As the riverdepth approaches top of bank conditions (point a), thecurve 'shifts' to the lower local minimum (point b),avoiding the depths between point a and b. For theflow of 4500 cfs, the lower depth has the minimumspecific energy Once again, a shift is necessary, fromthe upper minimum (point c) to the correspondingspecific energy for subcritical channel flow (point d).Both 'shifts' preserve energy and qualitatively includethe momentum transfer into the channel.


Figure 8 presents rising flows, where the momen-tum transfer is into the overbank. For 5500 cfs, thehigher critical depth is considered to separate subcrit-ical and supercritical flows. This means that flowwould be able to come out of bank and remain in

L supercritical conditions. This is not supported in prac-tice. Therefore, as the flow reaches the in-channelcritical depth (point a), a hydraulic jump is assumed,to a point lower than the extended specific energy(point b),and higher than the out of bank criticaldepth (point c). For a flow of 4500 cfs, the channelcritical depth is assumed to separate sub- and super-critical regimes. When the depth reaches bank fullconditions (point d), it will jump to a point above theout of bank critical depth (point e).

This proposed procedure represents a practical andhydraulic based method for handling multiple criticaldepths. In all cases, specific energy is conserved toprevent propagation of error to adjacent reaches. Thequestionable area immediately above the local maxi-mum of the curve is avoided. When included within aone-dimensional model, warnings are required todetermine when multiple critical depths occur andallow the user to further examine the flow character-istics. Though specific energy has been the mecha-nism of this paper, use of momentum produces similarresults.

SUMMARY

Multiple critical depths do occur in computationalmodels and are related to extreme changes in areaover small changes in depth. Modeling levees andmultiple main channel streams also can trigger theiroccurrence. There is some question whether the one-dimensional relationship, as currently applied, is'correct,' as the flow exceeds bank full conditions forlarge, instantaneous area changes. Equation (5)allows for simple prediction and location of multiplecritical depths through a geometric preprocessor foropen-channel flow models. A procedure is proposed towork through multiple critical depths by settingthe depth with the lower specific energy as the sub-critical-supercritical boundary. Depths immediatelyabove the suspect local maxima are avoided, and spe-cific energy is conserved.

Traver

5.5Depth (y) —— ft

a,

a)a)CLu

C,

7.5

U

Uaa

aU

U

Figure 8. Multiple Critical Depth Procedure — Rising Depth.


LiTERATURE CITED

Blalock, M. E. and T. W. Sturm, 1981. Minimum Specific Energy inCompound Open Channel. American Society of Civil Engineers,Journal of the Hydraulics Division 107:699-717.

Blalock, M. E. and T. W. Sturm, 1983. Closure to Minimum SpecificEnergy in Compound Open Channel. American Society of CivilEngineers, Journal of the Hydraulics Division 109:483-486.

Chaudry M. H., 1993. Open-Channel Flow. Prentice-Hall, Inc.,Englewood Cliffs, New Jersey.

Chaudry, M. H. and S. M. Bhallamudi, 1988. Computation of Criti-cal Depth in Symmetrical Open Channels. Journal of HydraulicResearch 26(4):377-396.

Chow, V. T., 1959. Open-Channel Hydraulics. McGraw-Hill Inc.,New York, New York.

Fread, D. L., 1988. The NWS Dambrk Model: Theoretical Back-groundftjser Documentation. National Weather Service, SilverSpring, Maryland.

Southworth, R. W. and S. L. Deleeuw, 1965. Digital Computationand Numerical Methods. McGraw-Hill Inc., New York, NewYork.

Traver, R. G., 1988. Transition Modeling of Unsteady One Dimen-sional Open Channel Flow Through the Subcritical-Supercriti-cal Interface. Dissertation presented to Pennsylvania StateUniversity, University Park, Pennsylvania.

Traver, R. G., 1993. Modeling Critical Depth in Open Channels. In:Hydraulic Engineering, H. W. Shea (Editor). Proceedings of the1993 National Conference, American Society of Civil Engineers,New York, New York.

U.S. Army Corps of Engineers, 1990. HEC-2 Water Surface ProfilesUser's Manual. The Hydrologic Engineering Center, Davis, Cali-fornia.

NOMENCLATURE

A = cross sectional area.d/dy = derivative.

g = gravitational acceleration.K = conveyance.n = Manning coefficient.

Q = discharge.= critical flow.

R = hydraulic radius.= specific energy.

y = depth of flow.a = energycorrection factor.

= constant for Mannings Equation.= individual subsection property.

— = cross section average.


modeling multiple critical depths

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